cours CT part1 - u-bordeaux.frtheo.ism.u-bordeaux.fr/~castet/doc4/Cours_CT_part1.pdf · Hˆ=! 1 2 " µ µ=1 2N #! Z A r A=1µ P # µ=1 2N #+ 1 r $=1µ$ 2N # µ

Theore&cal Chemistry

Frédéric Castet

frederic.castet@u-‐bordeaux.fr

Summary •  The Hartree-‐Fock-‐Roothaan method •  Pople and Dunning basis sets •  Semiempirical models •  Configura&on interac&on •  Möller-‐Plesset perturba&on theory •  Density func&onal theory •  Time-‐dependent DFT

Download material (lectures & prac&cals) at the address hHp://blake.ism.u-‐bordeaux1.fr/~castet/doc4.html

Douglas Hartree (1897-‐1958)

Vladimir Fock (1898-‐1974)

THE HARTREE-‐FOCK-‐ROOTHAAN METHOD

Clemens Roothaan (1918-‐)

H = ! 12"µ

µ=1

2N

# !ZArµA=1

P

#µ=1

2N

# +1rµ$$=1

2N

#µ<$

2N

# = hµ

µ=1

2N

# +1rµ$$=1

2N

#µ<$

2N

#

Electronic Hamiltonian for a molecule with 2N electrons and P nuclei

! =1(2N)!

"1(r1) "2 (r1) ... "2N (r1)"1(r2 ) "2 (r2 ) ... "2N (r2 )... ... ... ...

"1(r2N ) "2 (r2N ) ... "2N (r2N )


Single determinant wavefuncNon

!µ (rµ ) = "µ (rµ )# $µ (%µ )Spin-‐orbitals

Spin func&on α or β

Molecular orbital

HF: search for the best varia%onal wavefuncNon

x

y

z θ

ϕ

O

–e

r

r = r,!,"{ }dr = dV = r2 sin!drd!d"

VariaNonal principle

E = !H!dr " Eexact#1. Introduc&on of varia&onal parameters !"! #1…#M( )

2. Op&miza&on of the parameters

! E!"i

= 0 # i

Linear CombinaNon of Atomic Orbitals (Roothaan)

Basis set of known atomic func&ons

Expansion coefficients = varia%onal parameters

!i r( ) = Cpi"p r( )p=1

M

#

Minimiza%on of the total energy

HF: search for the best orbitals


Total energy

E = 2Iii=1

N

! + 2Jij "Kij( )j=1

N

!i=1

N

!

E = !H!dr"! =

1(2N)!

"1(r1) "2 (r1) ... "2N (r1)"1(r2 ) "2 (r2 ) ... "2N (r2 )... ... ... ...

"1(r2N ) "2 (r2N ) ... "2N (r2N )

Jij =!i*(rµ )!i (rµ )! j

*(r" )! j (r" )rµ"

# drµdr"

Ii = !i*(rµ )h!i (rµ )" drµ

HF integrals

Kij =!i*(rµ )!i (r" )! j

*(r" )! j (rµ )rµ"

# drµdr"

1-‐electron integrals

Coulomb integrals

Exchange integrals

Sum over N doubly occupied molecular orbitals


Roothaan expansion

Ii = !i*(rµ )h!i (rµ )" drµ = CpiCqihpq

q=1

M

#p=1

M

#

M2 integrals

Jij =!i*(rµ )!i (rµ )! j

*(r" )! j(r" )rµ"

# drµdr" = CpiCqiCrjCsj pq( rs)s=1

M

$r=1

M

$q=1

M

$p=1

M

$

pq( rs) =!p (rµ )!q (rµ )!r (r" )!s (r" )

rµ"drµ dr"#

Integrals in the AO basis

hpq = !p (rµ )h!q (rµ )" drµ

M4 integrals

pq rs( ) = qp rs( ) = pq sr( ) = qp sr( ) = rs pq( ) = sr pq( ) = rs qp( ) = sr qp( )


!i r( ) = Cpi "p r( )!p=1

M

#Atomic basis func%ons

Energy minimizaNon

E = 2Iii=1

N

! + 2Jij "Kij( )j=1

N

!i=1

N

! F(r)!i (r) = "i!i (r)Minimisa6on

N Fock equa%ons

F(rµ ) = h(rµ )+ 2Ji (rµ )! Ki (rµ )"#

$%

i=1

N

&

J i (rµ )! j (rµ ) =!i*(r" )!i (r" )rµ"

# dV"

$

%&&

'

())! j(rµ )

Fock operator

Ki (rµ )! j (rµ ) =!i*(r" )! j (r" )rµ"

# dV"

$

%&&

'

())!i (rµ )

Coulomb operator

Exchange operator

1-‐electron Fock operator

The Fock operator depends on its own solu%ons φ(r)

Itera%ve process un%l self-‐consistence

Describes electron µ in the mean electrosta6c field of the other electrons


Matrix form of the Fock equaNons

F(r)!i (r) = "i!i (r)

with

Fock matrix F

!

FC = "SC

Fpq = !p*" (r# )F(r# )!q (r# )dr# = hpq + Drs pq rs( )$ ps rq( )%& '(

s)

r)

Spq = !p*" (r# )!q (r# )dr#

Overlap matrix S

Dimension M x M

Elements

Dimension M x M

Elements

!i r( ) = Cpi"p r( )p=1

M

#

Dpq = niCpiCqii!

First-‐order density matrix D

Dimension M x M

Elements

Method of Linear Varia%ons

W. Ritz, J. Reine Angew. Math. 135, 1 (1909). See A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduc6on to Advanced Electronic Structure Theory, McGraw-‐Hill, New York, 1989.


General soluNon of the HF matrix equaNons for non orthogonal basis sets

By mul6plying by S–1/2 on each side

FC = SC!

!F

F!S"1/2 !S1/2 !C = S1/2 !S1/2C#

C = S!1/2 "C

S!1/2 "F"S!1/2! "## $## "S1/2 "C!"$ = S1/2C!#

!F !C !C

One seeks to obtain an eigenvalue equaNon of the form

Löwdin orthogonalizaNon P. O. Löwdin J. Chem. Phys. 1950, 18, 365

!! "F "C = "C #!C = eigenvectors of

!F !C = !C "


M basis func%ons ⇒ M MOs

ε MO

energy

Koopmans theorem: IP ≈ – ε

SoluNon of the HF equaNons

HOMO

LUMO

Virtual MOs

Occupied MOs


!(r) = ni" i

*(r)i=1

M

# "i(r)

Total electron density

!(r)space" dr = ni

i=1

M

# $i

*(r)$i(r)

space" dr = ni

i=1

M

# = N

Mulliken AO populaNons

!(r) = ni Cpiq=1

M

"p=1

M

" Cqi# p(r)

i=1

M

" #q(r) = Dpq# p

(r)#q(r)

q=1

M

"p=1

M

" Dpq = niCpiCqii=1

M

!

!(r)space" dr =N = DpqSpq

q=1

M

#p=1

M

# = Dpp + DpqSpqq$p

M

#%

&''

(

)**

p=1

M

# = Qp( )p=1

M

#

with

Qp = Dpp + DpqSpqq!p

M

" Electron popula%on in χp

nA = Qpp!A"

Mulliken atomic charges

!A = ZA " nA Net charge on atom A

ELECTRON DENSITY AND RELATED PROPERTIES

!µ = !

!µ!" dr = !

!µelec +

!µnuc( )!" dr

Dipole moment

!µelec = !eri

i" !!!!!!!!!!!!! !µnuc = eZA

!RA

A" !

!µelec = !2e "

i

*(r)# ri=1

occ

$ "i

*(r)dr

with and

!µelec = !2e CpiCqi "p (r)# r "q (r)dr

q=1

M

$p=1

M

$i=1

occ

$

= !e Dpq "p (r)# r "q (r)dr" #$$$ %$$$q=1

M

$p=1

M

$

ELECTRON DENSITY AND RELATED PROPERTIES

Dipole integrals

!µnuc = !

i

*(r)" eZA

!RA

A# !

i

*(r)dr = eZA

!RA

A#

VARIOUS TYPES OF BASIS SETS

!",n,l,mSTO r,#,$( ) =NYl,m #,$( )rn%1e%"r

Slater-‐type orbitals (STO)

STO are not efficient for evalua%ng the 3-‐ and 4-‐center integrals

!",n,l,mGTO r,#,$( ) =NYl,m #,$( )r2n%2%le%"r

2

The use of GTO ensures analy%cal solu%ons for all integrals appearing in the HF method

!i r( ) = Cpi "p r( )!p=1

M

#

Gaussian-‐type orbitals (GTO)

Atomic basis func%ons

PRACTICAL INTEREST OF GAUSSIAN FUNCTIONS

exp !"(r !A)2( )# exp !$(r !B)2( ) =Wexp !("+$)(r !P)2( )

Product of 2 Gaussian funcNons

The product of two Gaussian func6ons centered on A and B respec6vely is a Gaussian centered on the barycenter P of A and B

P = !A+"B!+"

W = exp !"#("+#)

A!B( )2$

%&

'

()with and

0

0,2

0,4

0,6

0,8

1

1,2

-‐20 -‐15 -‐10 -‐5 0 5 10 15 20

A=-‐4 α=1/50

B=+4 β=1/50

Product P=0, W=0.527

GAUSSIAN VS. SLATER FUNCTIONS: EXAMPLE OF HELIUM

He atom using a single STO

What is the best (varia%onal) STO to describe the ground state of He?

1s(r) = !3/2

"1/2exp(#! r)!!!$!!!!?

! r1,r2( ) = 121s(r1)1s(r2 ) "(#1)$(#2 )%$(#1)"(#2 )[ ]

E = 2I+ J = 2 1s(r)h1s(r)! dr! "## $##

+ 1s1s 1s1s( )! "# $#

Total energy

!2 2" 2! 5! 8

d Ed!

= 0"! =2716

=1.6875 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.5 1 1.5 2 2.5 3

1s o

rbita

l

Electron-nucleus distance r

E = !2 "278!

E = !2.84766!a.u.

He atom using a single GTO

What is the best (varia%onal) GTO to describe the ground state of He?

1s(r) = 2!"

#

$%

&

'(3/4

exp()!r2 )!!!*!!!!?

! r1,r2( ) = 121s(r1)1s(r2 ) "(#1)$(#2 )%$(#1)"(#2 )[ ]

E = 2I+ J = 2 1s(r)h1s(r)! dr+ 1s1s 1s1s( )

d Ed!

= 0"! = 0.766996

E = 3!" 8" 2#$

%&2!'

E = !2.300987!a.u.

Total energy

EGTO

>> ESTO


GTO much less efficient than STO!

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.5 1 1.5 2 2.5 3

1s o

rbita

l


Slater-‐type orbital (STO)

Gaussian-‐type orbital (GTO)

CUSP


CONTRACTED GAUSSIAN BASIS SETS

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.5 1 1.5 2 2.5 3

1s o

rbita

l


Slater-‐type orbital (STO)

Use of contracted Gaussian funcNons

1s(r) = aii=1

X

! " iG ! i

G =2"i#

$

%&

'

()3/4

exp(*"ir2 )

STO-‐2G

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.5 1 1.5 2 2.5 3

1s o

rbita

l


STO-‐3G Slater exponents α Contrac%on coefficients 0.6362421394D+01!0.1543289673D+00!0.1158922999D+01!0.5353281423D+00!0.3136497915D+00!0.4446345422D+00!

Slater exponents α Contrac%on coefficients 0.2432879285D+01!0.4301284983D+00!0.4330512863D+00!0.6789135305D+00!

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.5 1 1.5 2 2.5 3

1s o

rbita

l


Use of contracted Gaussian funcNons

-2.900

-2.850

-2.800

-2.750

-2.700

2 3 4 5 6

Tota

l SCF

-LCA

O e

nerg

y (a

.u.)

X in STO-XG

using a single STO function

Influence of the number of Gaussian contrac%ons on the total energy

CONTRACTED GAUSSIAN BASIS SETS

SCF-‐LCAO CALCULATION OF THE GROUND STATE OF THE HELIUM ATOM

PRACTICAL EXERCISE

Connect to the worksta6on bacon with your login and password ssh -Y [email protected]!Create a new directory named TP1 mkdir TP1!Go to the directory TP1 cd TP1!Copy the input file (data) used by the helium code cp /home/tp/Helium/Test/data .!Modify the data file as you need and save!kwrite data!Run the helium calcula6on!/home/tp/Helium/Src/helium < data > results!Open the result file (results) and note the relevant informa6on!kwrite results!

LOGIN AND PASSWORDS

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cours CT part1 - u-bordeaux.frtheo.ism.u-bordeaux.fr/~castet/doc4/Cours_CT_part1.pdf · Hˆ=! 1 2 " µ µ=1 2N #! Z A r A=1µ P # µ=1 2N #+ 1 r $=1µ$ 2N # µ